




«/• 









HOW TO HAKE Ml USE THEM 



B Y 




P\?ih$sor af Civ/'i Ei)ginee)yng, 
vers/ty of Pennsylvania. 




353 

Hsr 



PHILADELPHIA. 

Jos.M.Stoddartand CoMPANr. 

1881. 



LIBRARY OF CONGRESS. 



Shelf: Q'bJftb 



x. 



•* I 



UNITED STATES OF AMEBIC 



SRICi. 



V 



Working Drawings 



AND 



How to Make and Use Them. 



DESIGNED FOR 



INDUSTRIAL, TECHNICAL, NORMAL, AND THE HIGHER GRADE GRAM- 
MAR SCHOOL; ACADEMIES AND NIGHT SCHOOLS; AND 
ARTISANS DESIRING A KNOWLEDGE OF THE 
PRINCIPLES OF PATTERN AND 
TEMPLATE MAKING 



^ 




BY LEWIS M. HAUPT, 

Prof. Civil Engineering University of Pennsylvania. Late Director Franklin 

Institute Drawing School, Acting Assistant U. S. Coast and 

Geodetic Survey, &C, &*c, &*c. 



u U o 5. oY\ 



JOS. M. STODDARD & CO.: 

727 CHESTNUT STREET, PHILADELPHIA. 



1SS1. 



Entered according 


to the Act of Congress, 
LEWIS M. HAUPT, 


in the 


year 


1881, 


by 


In the office of the 


Librarian of Congress, 


at Washington, D. 


c. 



STEREOTYPED AND PRINTED 

BY INQUIRER P. & P. CO., 

LANCASTER, PA. 






PEEFACE. 

*. 

The large number of pupils of the public schools who 
are removed, by necessity, from their studies at the end 
of the grammar school course, and who are sent into 
mills, factories and shops as apprentices, seldom become 
more than a part of the machine which they are required 
to attend and operate, from lack of instruction in the 
one element which forms the -basis of all constructive 
development, namely, the science of drawing. 

It is true that, of late years, the art of drawing has 
been taught in the schools; that is, the manner of 
handling and using the necessary instruments to enable 
the student to copy from the flat or from models ; but 
all of this may be done mechanically, and is almost 
entirely destitute of intellectual culture. 

It is believed that there is but little in the present 
course of instruction that tends to develop the imagina- 
tion to such an extent as to enable a pupil to form a 
mental conception of an object from a mere verbal de- 
scription, and yet every new invention is but the fruit of 
such a process. It must first be clearly conceived in the 
mind, thence transferred to paper in the language of the 
artisan, and finally reproduced in substance out of ap- 
propriate materials. The inventor cannot convey, nor 

(i) 



11 PREFACE. 

the workman interpret his idea, unless both are familiar 
with the conventional language which must be employed 
to represent objects, not as they appear to, but as they 
do actually, exist, according to their true dimensions, 
relations and proportions. 

The present system of teaching drawing is useful only 
in so far as it cultivates the faculties of observing the 
form and position of objects, and the manual skill of 
representing them, requiring the exercise of the judg- 
ment and memoiy ; but is comparatively worthless for all 
practical purposes in the trades, except, perhaps, for the 
designer of free-hand patterns for tapestry, carvings and 
similar applications. The mere copying of pictures or 
models, or the construction of perspectives by rules-of- 
thumb, whose principles are not understood, is no more 
able to produce a draughtsman or artisan than would 
the copying of any number of sheets of music be able to 
make a musician, or the reproduction of hieroglyphics, 
a linguist. Any one who can handle a pencil may soon 
be taught to make a cop}^, although the characters so 
duplicated ma}' be unintelligible. 

To represent any object so that it may be constructed 
from the drawing, requires that it should be dissected 
and its several parts so projected on the plane of the 
paper that the artisan shall know just where to find 
them and what they represent ; in short, a knowledge of 
projections, scales, and the conventions used in working 
drawings must be understood. 

This is the missing link between theory and practice, 



PREFACE. HI 

which it is the effort of the author to introduce. With- 
out it all attempts to coordinate the industrial school 
features with our common school system must fail. 
The principles emplo}^ed are as simple and readily 
understood as those of elementary Geometry, upon 
which they are based ; and it is believed there is nothing 
in this Elementary Treatise on Working Drawings, and 
How to Hake and Use Them that is beyond the com- 
prehension of the average intellectual capacity found in 
the higher grades of our grammar schools. 

Whoever can make from his own working drawings a 
model of an object, should also, with appropriate tools 
and materials, be able to construct the object itself, or in 
other words, become a practical artisan so far as the 
general principles of framing and construction are con- 
cerned. 

In other countries numerous texts upon the subject 
are in use, but in America very few, and those only in 
our higher institutions of learning ; hence, one reason 
why many of our draughtsmen, designers and most suc- 
cessful artisans are foreigners. 

The plan of this work is to state the general principles 
involved in any theorem or problem, giving in the same 
connection its analysis, construction, and one or more 
of its numerous applications when practicable, thus fix- 
ing the principles much more efficiently than can be 
done by the ordinary methods of proceeding 

By this means it is hoped that the mental training re- 
sulting from a study of this subject, w T ill greatly assist 



IV PREFACE. 

in qualifying pupils for the more useful occupations in 
life, and reduce the number of graduates who are incom- 
petent to perform any other than clerical service, and 
who spend a large portion of their lives in seeking office. 

The application which may be made of such informa- 
tion is very extended. As a disciplinary study it is one 
of the first order, developing the conceptive faculties and 
enabling one to grasp an idea readily. It has its appli- 
cation in nearly all manufactured articles and in all con- 
structions and designs, in wood, iron, stone or other 
materials. It is used constantly by the engineer, archi- 
tect, builder, pattern-maker, iron or sheet metal-worker, 
stair-builder, stone-cutter, designer and many others. 
It is the basis of all perspective drawings, which are 
generally made by rule and without reason, and is essen- 
tial to a correct interpretation of all suggestions relat- 
ing to constructions of any kind. It is used to explain 
and reinforce verbal language, and should be so used 
whenever possible. 

One of its most important applications must not be 
overlooked. To the statistician as well as the merchant 
it is valuable as furnishing at a glance information which, 
if expressed in a mass of figures, would be unintelligible. 
It cannot be surpassed as a method of exhibiting rapidly 
the distribution of population, of products, of poverty or. 
wealth, of crime or morality, of vital, or in fact any sta- 
tistics which may be expressed numerically. To the 
physicist it is also particularly useful in investigations 
into the properties of molecular or mass physics, and 



PREFACE. V 

enables him to discover almost immediately many of the 
laws governing the motions of matter. 

Fluctuations of prices, in the market values of daily 
commodities, may be more intelligently expressed by this 
means than any other, and can be compared at a glance. 
In short, the number of intelligent and eminently practi- 
cal applications that may be made of projections is 
almost limitless. 

Its introduction into the grammar, normal and other 
schools would supplant a certain amount of mnemonical 
by rational and manual development, and would thus be 
a relief to a system already overtaxed with memorizing. 

In this brief treatise the consideration of the subject 
is limited to straight lines and planes ; but enough is 
given to enable the teacher to measure the capacity of 
the student, and to determine whether he has the ele- 
ments necessary to continue the subject with profit, in 
its application to curved lines, curved and warped sur- 
faces, and solids. 

Those students who may be incapable of developing 
the imagination to such an extent as readily to under- 
stand this part of the subject, should be allowed to stop 
here; but all who desire to become successful engineers 
or artisans must pursue the course still further, and 
take up the various subjects of intersections and develop- 
ments of surfaces and solids, as applied to pattern-mak- 
ing, machine-drawing, stone-cutting, and many other of 
the useful arts. These the author hopes to provide in 
subsequent numbers. This number is only intended to 



VI PREFACE. 

serve as a test or gauge by which the teacher may deter- 
mine which of his pupils may reasonably expect success 
in the many occupations in which these principles are 
employed, and also to serve as a basis for their further 
application to the trades and professions. 

The necessity for such instruction is becoming every 
year more urgent, in consequence of the abolition of the 
system of apprenticeships. The author knows of no 
other means that will so rapidly and cheaply meet this 
difficulty than a thorough ground-work in the principles 
of projections as applied to making and reading working 
drawings. L. M. H. 



SUGGESTIONS TO TEACHERS. 



As this subject may be new to many teachers, the 
author has given, in addition to the ordinary projections, 
as in the figures lettered (a), a perspective view of the 
objects in space, to assist the imagination. The paper 
planes will also be found a great help. The drawings of 
the problems should be made upon them unfolded ; they 
may then be turned up so as to form a solid angle, and 
the position of the object (point, line or plane,) be indi- 
cated by holding a pencil, pointer, or card, in the 
proper position. 

Frequent practice of this kind will render the student 
very expert at reading the drawings. 

The teacher will find a hinged blackboard of two 
leaves very useful, and a card or thin board with a hole 
in it, through which a needle or wire may be placed, will 
enable him to illustrate most of the problems by a model. 

Students should also be required to prepare drawings 
of the problems given in the text, assuming the parts 
in different positions, and to apply the principles to 
other objects than those suggested. 

It will be found beneficial to require the student to 

give all the principles involved in the solution of any 

problem, and to state where they are applied. 

(vii) * 



Vlll SUGGESTIONS TO TEACHERS. 

By attention to these few hints, the author hopes 
that the teacher who may take up this subject for the 
first time will find it simple, pleasant and instructive. 

L. M. H. 



CONTENTS. 



INTRODUCTION. 

Section. 

1. Drawing, defined and classified, . . . 

2. Form and color as general characteristics, . 

3. A Point defined ; how determined ; its locus, 

4. A Line defined ; straight, curved and broken lines, 

5. An Angle defined ; plain, spherical, 

6. Perpendiculars ; right and oblique angles, . 

7. Directions and relative positions of lines, 

8. Parallels, . ■ . . . 

9-15. Lines, oblique, vertical, horizontal, tangent, secant, diago 

nal, segments of 

16. A Plane, defined, . . 

17. A plane, determined, 

18. Right line and plane, 

19. Right line and plane, relative positions of . 

20. Two Planes, intersection of, 

21. Two Planes, relative positions of, 

22. Diedral Angle, faces, edge of, measure of, 

23. Curved Lines, classified, single and double, 

24. Plane Curves, . . . 

25. The Circumference, defined, . 

26. An Arc, . . . . 

27. The Parabola, defined, 

28. The Ellipse, ..... 

29. The Hyperbola, , 

30. The Oval, 

31. The Cycloid, .... 

32. The Epi- and Hypo-Cycloids, . 

33. The Involute, . . . 

34. The E volute. .... 

(ix) 



PAGE 
. 3 

3 
. 4 
4 
, 5 
5 
5 
5 

6 
6 

7 
7 
7 
7 
7 
7 
8 
8 
8 
8 

8 
S 
8 
8 
9 
9 
9 
9 



CONTENTS. 



Section. 

35. The Spiral, 

36. The Catenary, . . . . . " , 

37. A reverse curve, ..... 

38. A compound curve, ..... 

39. Ogees, cusps, etc., . . . 

40. Curves op Double Curvature, the Helix, . 

41. The pitch of a thread defined, 

CHAPTEK I. 

42. Working drawings defined, 
43." The Scale defined and its use explained, . 

44. Scales expressed as ratios, formula, 

45. Quantities must be reduced to same denomination 

46. Manner of constructing and marking a scale, 



PROJECTIONS. 

47. Definition, . . . .12 

48. The essential data, 12 

49. The planes of projection, 13 

50. The ground line, 13 

51. The angles, 13 

52. Position of the point of sight, 13 

53. Protectants or perpendiculars, . . . . . .14 

54. Nomenclature of a point, ...... 14 

55. Projections of a point in space, .14 

56. Exercises, 15 

57. Projections of a point situated in the planes of projection, . 16 

58. To change the scale, . . . . . . . 16 

59-61. Points in different angles, 17 

62. Revolution of the vertical plane, . . . . . 18 

63. Position of points after revolution, . . . . .18 

64. Revolution of a point about a line or axis, IS 

65. To find the revolved position of a point, . . . .19 
6Q. Analysis of above problem, ...... 19 

67. To determine the radius of the path of a point when the pro- 

jections of the point are given, . . . .19 

68. Construction of the problem, ...... 20 

69. General rule to find the revolved position of a point, . . 20 
70-74. The positions of the projection of points in different 

angles, 21, 22 



PAGE 
. 9 

9 
. 10 

10 
. 10 

10 
. 10 



11 
11 
11 

12 
12 



CONTENTS. xi 

Section. page 

75. Both projections of the same point are in the same straight 

line perpendicular to the ground line, ... 22 

76-80. Geometrical principles involved, 23 

81-82. Distance of a point from either plane of projection, . 23, 24 
83. To determine a point in space from its projections, . . 24 

84-85. Exercises, 24, 25 

86-88. Conventionalities, 25, 26 

CHAPTER II. 

THE RIGHT LINE. 

89-93. Axioms and geometrical principles, . . . .27 
94. To represent a straight line by its projections, . . 27 

91. The point in which an oblique line pierces a plane is a point 

of its projection, 28 

96. To assume a point on a given line, ..... 28 
97-105. Positions of a line in space, . . . . . 29, 32 
106. To determine the line from its projections, ... 32 
107-108. Exercises, 32, 33 

109. To find the point in which an oblique line pierces either 

plane of projection, . ...... 33 

110. Eule, 34 

111. Same, when line is parallel to planes of projection, . 34 

112. If two lines intersect at a point, their projections must pass 

through the projections of the point, . . . .35 

113. Converse of above (112), 35 

114. To draw a line through a given point on a line, . . 36 

115. To draw two lines through a given point, ... 36 
1.16. Exercises, .... 36 

CHAPTER III. 

PLANES. 

117. Planes determined, 37 

118. Traces defined, 37 

119. The traces of an oblique plane must intersect each other, if 

at all, on the ground line, 37 

120. To represent a plane in any position, . . . .38 

121. Applications to a block of wood or stone, ... 39 

122. Plan, elevation, and section, 40 

123. To represent a hollow box, 40 



xii CONTENTS. 

Section. page 

124. To determine the true sizes of the patterns, . . .41 

125. Manner of indicating dimensions, 41 

126. To represent a brick and stone wall, 41 

127. To represent a mortise and tenon joint in wood, . . 42 

128. To find the revolved position of a point on a side plane, . 43 

CHAPTEK IV. 

129. To find the true length of an oblique line, ... 44 

130-131. Other methods, 45 

132-134. To assume a line in any oblique plane, ... 45 

135. To pass a plane through three given points, . . .47 

136. To determine the plane of two lines, . . . .48 

137. If a system of lines be parallel, their projections must be 

parallel, . . 48 

138. To pass a plane through one given line and parallel to 

another, 48 

139. To pass a plane through a right line and a point. . .48 

TWO PLANES. 

140. To find the intersection of two oblique planes, . . 49 

141. To find the angle which the line of intersection makes with 

either plane of projection, . . . . . .49 

142. The angle which a line makes with a plane is equal to that 

made with its own projection on that plane, . . 50 
143-145. Special cases of intersections of planes, . . .50 

LINE AND PLANE. 

146. To find the point in which a given line pierces a given 

plane, 52 

147. To find a point on a plane when either of its projections 

is given, ......... 53 

PRACTICAL APPLICATION. 

148. Drawings and estimates for a bird-house, dog-kennel, or 

similar structure, 53 



HSTRODUOTOBY. 



DRAWING DEFINED. 

POINTS AND LINES CLASSIFIED AND DEFINED. 

1. Drawing is the art of representing objects or ideals 
(a) as they exist or (b) appear to exist. 

(a) The representation of objects as they exist, show- 
ing their forms, sizes and positions, is included in those 
subdivisions of the art known as Orthographic and 
Isometrical Projections. 

(b) The representations of objects as they appear to 
exist is known as Scenographic Projections, or Linear 
Perspective. 

To draw correctly, therefore, a knowledge of projec- 
tions must first be acquired as a foundation for subse- 
quent work. 

2. All objects possess two general characteristics, 
form and color, which impress themselves upon the 
minds of all rational, seeing beings. The color, which 
gives expression to objects, is limited by and contained 
within the form. The student's first care should there- 
fore be to represent the form correctly; and since the 
outlines of all objects are composed of Jines, either 
straight or curved, or both combined, he must neces- 

(3) 



4 INTRODUCTORY. 

sarily devote some time to the consideration of these 
elements in their order, before taking up the subject of 
projections. 

POINTS, LINES AND ANGLES. 

POINTS. 

3. The origin of all lines is the point, which, mathe- 
matically considered, is a mere ideal having no material 
existence, and consequently " neither length, breadth 
nor thickness, bat position only." The points used in 
drawings have, however, a visible existence, and are rep- 
resented by mere dots, so small indeed, that their 
actual dimensions are supposed to be incapable of being 
measured. They are used simply to indicate positions. 

A point is given or determined when its position 
with reference to other points or lines is known. 

The intersection of two lines is always a point, whose 
position is determined when that of the lines is known. 

lines. 
If a point be put in motion it will generate a line 
called the path or locus of the point. 

4. If the motion of the point be always in the same 
direction, the path will be a right or straight line; if 
it continually change its direction, in accordance with 
some law, it will be a curved line. A curved line 
may also be defined as one in which no three consecutive 
points lie in the same direction. Thus a line is a 
geometrical •magnitude composed of a succession of 
points lying in the same or different directions. 



INTRODUCTORY. 

A broken line is one composed of portions of straight 
lines. 

ANGLES. 

5. An angle is the portion of space included between 
two lines or planes which intersect. If the lines are 
straight, the included space is a plane angle ; if arcs of 
spheres, a spherical angle. 

6. Two right lines are perpendicular or " square'' to 
each other when the angles around their point of inter- 
section are all equal.* The angles are then right 
angles. An angle less than any one of these is said to 
be acute ; one greater, obtuse. 

DIRECTIONS AND RELATIVE POSITIONS OF 

LINES. 

7. Every straight line has two directions, as that from 

left to right, or from bottom to top, as from A B 

A toBj or the reverse, as from B to A. 

8. Two right lines are parallel, when they both lie 
in the same direction. The angle between them is there- 
fore zero.f Two concentric circumferences, or those 
described from the same centre, are parallel, as well as 
right lines lying in the same direction ; for their tangents 
(12) at corresponding points will be parallel. 

*It is not necessary that two lines should lie in the same plane nor 
intersect to be perpendicular, for they are so if 'their directions in space 
are at right angles to each other.. 

f This fact is important, as it is the basis of the method of finding the 
vanishing point of any system of parallel lines in perspective. 



6 INTRODUCTORY. 

9. An oblique line is one which makes any angle with 
another line different from zero or a right angle. It 
may have any position, therefore, between the parallel 
and the perpendicular. These positions may be easily 
illustrated by revolving a line in a plane about one of its 
extremities, and noting the angles which it makes with 
its first position. 

10. A vertical line is one which is perpendicular to 
the surface of water at rest, or which is parallel to a 
plumb line. 

11. A horizontal line is one which is at right angles 
or perpendicular to a vertical line. A horizontal line is 
therefore parallel to the surface of water at rest, or to 
the earth's sensible horizon. 

12. A tangent is a line which simply touches another 
at an} r point, but does not cut it. 

13. A secant is a line which cuts another in one or 
more points. 

14. A diagonal is a line joining non-adjacent angles 
in any figure. 

15. The segments of a line are the portions into which 
it is divided by any cutting line or point. 

A given line, or surface, is one whose position is 
known or assumed. 

planes. 

16. A plane is a surface generated by a straight line, 
moving so as to touch two parallel straight lines, or so 
as to revolve in a direction perpendicular to a given 
straight line. Hence, if any two points of a plane be 



INTRODUCTORY. 7 

joined by a right line, that line will be wholly within 
the plane. 

17. A plane is determined — that is, fixed in position — 
by the conditions, that it shall pass through two straight 
lines which may intersect, or be parallel; or through a 
right line and a point; or through three points not in the 
same straight line, or through a given point and be per- 
pendicular to a given line ; or through a given line and 
be perpendicular to a given plane, and by many other 
reasonable conditions. 

RIGHT LINE AND PLANE. 

18. The relative positions of a right line and a plane 
are similar to those already given for two right lines. 

19. A line is said to pierce a plane, and a plane to cut 
a line. When the line passes through or pierces the 
plane, the intersection will be a point ; but when the 
'plane passes through the line, the line lies wholly within 
the plane 

TWO PLANES. 

20. The intersection of two planes is always a right 
line. 

21. The relative positions of two planes are similar to 
those of two right lines — that is, they may be perpen- 
dicular, parallel, or oblique to each other ; secant, diag- 
onal, horizontal or vertical. (See Page 5.) 

22. The angle formed by two planes which intersect, 
is called a diedral angle ; the planes are the fjlCES, and 
the line of intersection the edge, or arris. 

The diedral angle is measured by, or equal to, the 



8 INTRODUCTORY. 

plane angle formed by intersecting the faces by a plane 
perpendicular to the edge at any point. 

CURVED LINES. 

23. Curved lines may be divided into two general 
classes, as those of single and those of double curva- 
ture, according as the generating point moves in a 
plane, or in space. 

PLANE CURVES. 

24. A PLANE CURVE, Or CURVE OF SINGLE CURVATURE, is 

one, every point of which lies in the same plane. The 
principal curves of single curvature are : 

25. The circumference, which is generated by a point 
moving in a plane 3 so as to remain at the same distance 
from a fixed point called the centre. 

26. An arc is any portion of a circumference. (Care 
must be taken to distinguish between a circle, which is 
an area, having two dimensions, and its circumference, 
or bounding line, having but one.) 

27. The Parabola, every point of which is equally distant 
from an assumed point, or focus, and a straight line not passing 
through the point. 

28. The Ellipse, every point of which is so situated that the 
sum of its distances from two fixed points, or foci, is equal to a 
given straight line. 

29. The Hyperbola, every point of which is so situated, that 
the difference of its distances from the tw T o foci is equal to a given 
straight line. 

[Note. — These four curves constitute the conic sections, since 
they may all he cut from the surface of a cone by a plane. They 
may be consiructed from the definitions above given, as will be 
shown hereafter in the application of these principles.] 

30. The Oval, a plane curve composed of arcs of circles, which 



INTRODUCTORY. 9 

are tangent to each other, two and two, and closely resembling 
the ellipse in appearance. 

CYCLOIDS. 

31. The Cycloid is the path described by any point in the plane 
of a circle, which is rolled along its tangent. The term is gene- 
rally limited to the curve generated by a point on the circumfer- 
ence. If the point is within the circumference, the curve is a 
prolate cycloid; if without, a curtate or contracted cycloid. 

32. The Epicycloid is the curve generated by a point on the 
circumference of a circle which rolls around another. If the'gen- 
erating circumference moves upon the outside of the fundamental 
circle, the epicycloid is external; if on the inside, it is internal, or 
more generally an hypocycloid. 

(These curves are important as forming the outlines of toothed 
wheels, and of rack and pinion gearings.) 

33. The Involute is a plane curve generated by the end of a 
flexible line which is unwound from the circumference of a circle 
or other curve. 

34. The Evolute is the curve about which the line is wound. 

35. If more than one turn be made, the line generated will be a 
Spiral. It may be constructed from the above definition of an 
involute, by drawing tangents to an evolute of any assumed form 
(but generally a circumference is taken), and laying off on these 
tangents the lengths of the evolute, from the origin of the spiral 
to the respective tangent points. This construction may be 
readily illustrated practically by wrapping a string around a cir- 
cular block or wheel, and placing a pencil or piece of chalk at the 
loose end of the string, the other being fastened to the block. If 
now the pencil point be moved away from the block or evolute, as 
the string is unwound keeping it stretched tightly, the curve de- 
scribed will be a spiral. The term spiral is often incorrectly used 
for a curve of double curvature, which it is not, as of "spiral " 
stairs or a "spiral " screw thread, etc. These curves are helices, 
as will appear presently. (40.) A steel watch-spring is a familiar 
instance of a spiral. 

36. The Catenary is a plane curve formed by suspending a 
flexible chord or chain, uniformly loaded, from two fixed points 
of support. 

1* 



10 INTRODUCTORY. 

37. A reverse curve is one composed of two simple curves, 
which, lie on opposite sides of a common tangent, thus v^_/ N ; 
in the trades it is called an ogee, or " O. G." 

38. A compound curve is one composed of two simple curves 
of different radii lying on the same side of a common tangent, as 
in an oval. 

39. There are many other plane curves having definite names 
and properties, as the Ogees, or cyma recta and cyma reversa, 
Ovolos, Cusps, Smoids, etc., which it is unnecessary to consider in 
this connection. 

CURVES OF DOUBLE CURVATURE 

40. The principal curve of this class is the Helix, which is 
generated by a point moving uniformly in the direction of a 
straight line, while at the same time it revolves so as always to re- 
main at a constant distance from it. (The curve can be illustrated 
by wrapping a right-angled triangle cut from a piece of paper 
around a cylinder. It is the same as the edge of a screw-thread 
or hand-rail to a winding stair-case.) 

41. The distance through which the point moves in making one 
revolution, measured parallel to the straight line or axis, is called 
the pitch. 

(The number of curves of this class is practically infinite, but 
this is the only one which it is necessary to study in this con- 
nection.) 



CHAPTER I. 

WORKING DRAWINGS. 

42. Working drawings are projections (47) of the 
objects to be represented, or made, drawn to a scale. 

SCALES. 

43. A scale is an instrument for determining the ratio 
of the object to the drawing. Thus, if the drawing is 
made so small that one inch on it represents one foot of 
object, the scale is then one foot (or twelve inches) to 
one inch, or ^ 

As the object to be drawn has generally a definite size, 
whilst the drawing ma}^ be made of any convenient 
magnitude, it is better to take the former as the divisor 
or denominator of the fraction expressing the ratio, and 
the corresponding dimension of the drawing as the 
numerator. 

44. Thus we have the ratio of the object to its 

1 . Drawing D. 

drawing, — s or = 

- Object 0. 

The scale of a drawing or map is then obtained by 

dividing the second quantity or drawing, by the first or 

object. Thus, a map drawn to a scale of one mile to an 

inch, evidently means one mile of ground to one inch 

of paper, or 63360 inches of ground to one inch of the 

(11) 



12 WORKING DRAWINGS 

drawing — that is, -g^eiF— so f° ur ^ ee t to one inch means 
that 48 inches of the object or model are represented by 
one inch of the drawing ; hence, the scale is ^L. This 
scale is frequently erroneously called "J inch to 1 foot," 
which is just the reverse of that intended. 

45. The quantities used in expressing the ratio must 
always be reduced to the same denomination. 

46. Instead of expressing the scale fractionally, it 
may be drawn out on the paper, and marked so that its 
divisions shall represent the proper size of the object. 
Thus, a scale of 1 foot to 1 inch or y 1 ^ may be represented 
by a bar or line divided into inches, but marked feet ; 
thus 



1 o 1 2 Ft. 

The left hand inch may be subdivided into tenths, 
twelfths, sixteenths, or any other suitable fraction.* 

PROJECTIONS. 

47. The word project means, literally, to throw down 
or upon ; hence the projection of an object is the repre- 
sentation of it made b\ r throwing it vertically down or 
upon the plane on which the drawing is to be made. 

48. Three things are essential in all projections : they 
are, the position of the point from which the object is 
seen, called the point of sight; the position of the object 
in space; and the position of the planes upon which the 
drawing is to be made, 

* Mechanics generally use the duodecimal scales; engineers, the 
decimal. 



AND HOW TO MAKE AND USE THEM. 13 

THE PLANES OF PROJECTION. 

49. To represent the three dimensions of any solid 
body there mnst be at least two planes, intersecting at 
right angles, upon one of which the length and breadth 
can be measured off, and upon the other the height ; but 
since these two planes do not show a section of the 
body, it is necessary to use a third, at right angles to 
the intersection of the others, as in the three adjacent 
faces forming the solid angle of a cube. See Fig. 1. 
These are called the planes of projection, and are des- 
ignated as the horizontal, ground or (H) plane ; the ver-< 
tical, wall, or (Y) plane, and the side, section, or (S) 
plane. They are also called coordinate planes. 

50. The line of intersection A B, of H and V, is 
called the Ground Line. 

51. The diedral angles formed by these H and Y 
planes produced are designated as indicated on the 
figure, that above H and in front of Y being the first 
angle, and so on around as in Fig. 1. 

These coordinate planes are supposed to be of in- 
definite extent, so that any object in space will be found 
either in some one of the four angles or on the planes 
which separate them; and as the size of the object is not 
affected by its distance from the planes, it will be found 
more convenient to imagine them placed so close to the 
object that it may rest upon them. 

POINTS* OF SIGHT. 

52. In all working drawings the point of sight is 
situated in lines perpendicular to the planes of projec- 



14 WORKING DRAWINGS 

tion, and at an infinite distance from them ; and since 
there are three such planes, there must be three different 
positions for the points of sight, *one for horizontal pro- 
jections, another for vertical, and the third for side- 
views. 

53. From the positions and distance of the point of 
sight it follows that all visual rajs drawn through 
points of the object to either plane will be parallel to 
each other and perpendicular to that plane. Such rays 
are called the projecting lines or projectants of the 
object, and the points in which they pierce the planes of 
projection are the projections of the points through 
which they pass. 

nomenclature of a point. 

54. To distinguish the vertical from the horizontal 
projections, it is necessary to designate them by some 
characteristic signs ; hence for a point in space the cap- 
ital letters are used, as P; while for its horizontal projec- 
tion the corresponding small letter is taken, as p ; and 
for its vertical projection the same small letter accented, 
as p' (called p prime). The revolved position is gener- 
ally designated by the same small letter with the 
"second" mark, as p'\ 

THE PROJECTIONS OF A POINT IN SPACE. 

55. Problem. — To project any given point we have 
only to draw through it its projectants, and find where 
they pierce the planes of projection. These points will 
be the projections of the given point, the one in H being 



AND HOW TO USE THEM. 15 

the horizontal projection, and the one in Y the vertical. 
Thus if a point (P), Fig 2, be situated in the first 
angle, at a distance of three-fourths of an inch above 
H, three-fourths of an inch in front of V, and half an 
inch to the left of S, we have only to lay off on BA 
one-half inch from B ; on c p three-fourths of an inch, 
and on p P three-fourths of an inch, and the given 
position of the point will be represented. 

Note — Pupils should draw the positions of a sufficient 
number of assumed points to enable them to conceive 
readily of their positions in space, and should hold the 
end of a pencil or pointer in the position of the imag- 
inary point in space to indicate its place. 

exercises. 
56. In the following exercises th$ coordinates of sev- 
eral points are assumed for practice. The distances 
given are the actual distances measured perpendicular to 
the planes of projection, as indicated under the letters 
S, V and H, respectively. In the following assumed 
positions, all distances measured to the right from S, to 
the front from V, and upward from H, are plus ( + ), 
or positive; all distances measured to the left from S, 
to the rear from Y, or downward from H, are minus 
( — ), or negative; hence the coordinates of points in 
the first angle and to the right of S will all be positive ; 
for those to the left, the S distances will be negative ; 
for those in the second angle, distances from Y will be 
negative ; for those in the third, both the Y and H dis- 
tances will be negative ; and for those in fourth angle, 



16 



WORKING DRAWINGS 



distances from V will be positive, and those from H 
negative. 

CO-ORDINATES OF POINTS TO BE PLATTED (NATURAL SCALE). 



Name of Point. 



Fig. 2. 



Fig. 2. 



P 

Q. .. 

K. . . . . 

S 

T 

U 

P 

p\... 
c 



Distance 
from S. 



- J inch. 



— 1 
— 1 

-n 

—2 



Distance 


Distance 


from Y. 


from H. 


f inch. 


5 inch. 


1 " 

1 a 
— 2 

-H" 

3 <i 
4 


1 a 
4 

— 1 " 


H" 

3 a 
4 

" 

" 


2 " 

" 
j, << 

o 2 " 



In what angle ? 



First, etc., to be 
filled out by pu- 
pil. 



On H. 
On V. 
On Ground Line 



57. It will be seen from the last three cases that, if the 
points be in the planes of projection or at their intersec- 
tions, one or both of their projections will be on the 
ground line, and theft positions will be determined in the 
same manner as when in space. 

58. If the above co-ordinates be multiplied by twelve, 
the number of inches in one foot, the numbers in the 
table will represent the distances of the points from the 
planes in feet, instead of inches ; but the same drawing- 
may still be used by changing the scale, or assuming it 
to be ^2 the full size ; this is done by marking the inch 
divisions on the scale to represent feet, thus : 

Scale 1 ft. to 1 in., or T V 



Ft.i % % % o l 2 

In the same manner any other scale may be applied, as 
2 feet to \ ,f or J- ¥ ; three feet, or one yard, to one inch, 



or 



365 



etc. 



and how to use them. 17 

Fig. 2. 

59. If the point be in the second angle, as at P 2 , its 
projections will be p 2 and p r 2 . 

60. If the point be in the third angle, its vertical pro- 
jection p' 3 will be on that part of V below the ground 
line, and its horizontal projection p 3 on that part of H 
behind AB. 

61. In the fourth angle, the vertical projection p' 4 is 
below the ground line, and the horizontal projection p 4 
is in front. 



18 



WORKING DRAWINGS 



REVOLVED POSITIONS. 



A 



REVOLUTION OF THE PLANES OF PROJECTION. 

62. Since it is practically impossible to make drawings 
upon three sheets of paper placed at right angles to each 
other, it becomes necessary to turn the vertical planes 
down so as to coincide with the horizontal. For this 
purpose the side plane is revolved 90° to the right, about 
the line BC (Fig. 2) as an axis, until it lies in the verti- 
cal plane produced. The vertical plane is then turned 
down backward through n 
the second angle until it 
coincides with the horizon- 
tal plane. Thus the three 
planes of projection are rep- 
resented on the same sheet 
of paper. The ground line 
AB separates the horizontal or lower portion from the 
vertical or upper portion, whilst the side plane will lie to 
the right of the vertical line BC, as above. 

63. All the points projected on the vertical planes 
will remain in them, and be found in the same relative 
positions with reference to the lines AB and BC, as 
before the revolution. Hence, it follows that all hori- 
zontal projections of objects in the first angle ivill be 
below the ground line, all vertical above, and all side 
projections, looking from the left, will be above AB 
and to the right of B C. 

REVOLUTION OF POINTS ABOUT AN AXIS. 

64. A point is said to revolve about a right line as 



V 

Vertical 
Plane. 


s 

Side 
Plane. 

i 


Horizontal 
Plane. 

H 


B 



AND HOW TO USE THEM. 19 

an axis, when it moves in a plane which is perpendicular 
to the line, and remains at a constant distance from it. 
The path of the point will then be a circumference, of 
which the radius is the distance of the point from the 
axis. The axis may be in any position — as horizontal, 
vertical, or oblique. 

65. Problem. — To find the revolved position of a 
point about a given axis. 

Note. — The position of a given point with reference 
to a given line is determined when we know both its 
distance and direction from some point of the line. 

66. Analysis. — The distance of a point from the axis 
is always measured by the perpendicular from the point 
to the axis ; this distance is the radius of the circle of 
revolution; and since this radius must remain perpen- 
dicular to the axis, the direction to any new position of 
the generating point must always be perpendicular to 
the axis. Hence to find the revolved position of a 
point, through the centre, draw a line perpendicular to 
the axis, and make it equal to the radius of the circle 
described by the generating point. 

67. To determine the length of this radius and the 
position of the centre when the point is given by its pro- 
jections. 

The distance of the point, from an axis in the hori- 
zontal plane, is the hypothenuse of a right angled 
triangle, the height of which is the distance of the point 
above the horizontal plane, or the distance of its vertical 
projection above the ground line, and the base is the 



20 WORKING DRAWINGS 

distance from the horizontal projection of the point to 
the axis. 

68. Construction. — Thus, in Fig. 3 and (b), let P be a 
point in the first angle ; let M N be an axis lying in the 
horizontal plane, about which it is proposed to revolve 
the point P unti] it falls upon said plane. The hori- 
zontal projection of P is p, and the vertical projection 
p'. The radius of the path Pp /r is c P, which is evi- 
dently the hypothenuse of the right angled triangle 
cPp, whose plane is perpendicular to MN; and therefore 
cP must be perpendicular to MN.* Hence to find c, the 
centre, drawn through p, which is always given, a line 
perpendicular to the axis ; the point of intersection c 
will be the centre ; and to find the radius draw any- 
where two straight lines at right angles to each other, as 
cp and pP, Fig. 3 (a), and lay off on them the lengths 
of the base and altitude of the triangle cpP, as given by 
the lines c p and P p or its equal p'd. Fig. 3. The line 
joining the points thus determined will be the hypothe- 
nuse or radius required. 

69. Rule. — Hence we have this general rule for 

FINDING THE REVOLVED POSITION OF A POINT ABOUT AN 

axis in H. Through the horizontal projection of the 
point draw a, straight line perpendicular to the axis; 
lay off on it, from the axis, a distance equal to the hypoth- 
enuse of a right angled triangle, the base of which is 
equal to the distance of the horizontal projection of the 

* For if a plane is perpendicular to a line, any line in that plane will 
be perpendicular to the line. 



AND HOW TO USE THEM. 21 

point from the axis, and the altitude, the distance of the 
vertical 'projection of the point from the ground line. 

If the axis be in V, the same rule will apply by inter- 
chaiiffing the words vertical and horizontal. 

70. TO FIND THE POSITION OF THE PROJECTIONS OF POINTS 
AFTER THE REVOLUTION OF Y INTO H. (See 63.) 

To apply this rule to the revolution of Y into H, Fig. 
2, we see that the axis AB must pass through the hori- 
zontal projection of all points in Y, and hence the base 
of the triangle vanishes or becomes zero (0), whilst the 
hypothenuse and altitude coincide or are equal. The 
radius is therefore the distance from the point in Y to 
the ground line, and the revolved position of p' will be 
found at p", in the perpendicular to AB at c, and at a 
distance from it, p ;/ c equal to p'c. 

71. Hence, if the point be in the first angle, as P, its 
vertical projection must be above the ground line, and 
its horizontal projection below. 

72. If in the second angle, as P 2 (Figs. 2 and 4), since 
all vertical projections above the horizontal plane are 
found above the ground line, and since after revolution 
that part of Y coincides with that part of H which is be- 
hind the ground line, both projections will be on the same 
side or above the ground line, and can only be distin- 
guished by their letters (see 54). 

73. If in the third angle, as P 3 (Figs. 2 and 4), the rear 
portion of H after revolution will be above AB, and the 
lower portion of Y below ; hence, the horizontal projec- 
tion will be above, and the vertical below the ground line 
— just the reverse of the point in the first angle. 



22 WORKING DRAWINGS 

74. If in the fourth angle, as P 4 , it will be the reverse 
of that in the second, or both projections will fall below 
the ground line. (See Figs. 2 and 4.) 

Note. — Since the side plane is only useful when sec- 
tions are desired, it may be omitted for the present, and 
the attention be confined only to the planes V and 
H, which are now supposed to lie in the same plane, but 
to be separated by the ground line as represented in 
Fio-. 4. 

75. Theorem. — The two projections of a point are in 
the same straight line perpendicular to the ground line. 

Analysis.— If a plane be passed through any point 
(P) in space, perpendicular to the ground line, it will 
contain the projecting lines of the point, and will cut 
the planes of projection in two lines, which are perpen- 
dicular to the ground line at the same point. Now, when 
Y is revolved into H, these last two lines will lie in the 
same plane and be perpendicular to the ground line at 
the same point ; hence, they must coincide or form one 
and the same straight line. 

Construction. — Fig. 2. Let P be the point. The 
plane formed by the lines Pp and Pp' is perpendicular 
to AB ; hence, AB is perpendicular to the lines cp and 
cp', which lie in the plane Pc. But when cp' is revolved 
it will coincide with cp", which is also perpendicular to 
AB, and therefore pc and cp" form one straight line 
upon which the projections p and p' or p 7/ are found ; 
hence, both projections of the same point must always 
lie in the same straight line perpendicular to the ground 
line. 



AND HOW TO USE THEM. 23 

Principles of Geometry involved. Fig. 2. 

76. 1. Two straight lines intersecting at a point P, 
determine a plane p'Pp. 

' If. 2. If a straight line (Pp or Pp') is perpendieular 
to a plane (H or V), any plane passed through that line 
must be perpendicular to that plane ; hence, p'Ppc is 
perpendicular to both H and V at the same time. 

T8. 3. If a plane (p'Ppc) is perpendicular to two 
others (H and Y) which intersect, it must be perpendic- 
ular to their intersection (AB). 

19. 4. If a plane (p'Ppc) is perpendicular .to a line 
(AB), any lines (cp ; or cp) lying in that plane, will be 
perpendicular to the line (AB). 

80. 5. If a point p' revolve about a line (AB) as an 
axis, it must move in a plane perpendicular to the axis. 

THE POINT IN SPACE. 

81. Theorem. — The distance of a point from H is 
measured by the distance of its vertical projection from 
the ground line, and the distance of a point from V is 
measured by the distance of its horizontal projection 
from the same line. 

Since the figure cp'Pp (Fig. 2) is a rectangle, whose 
opposite sides are equal and parallel, we have cp=p'P, 
or the distance of the horizontal projection from the 
ground line, is equal to the distance of P in front of Y, 
and since Pp=p'c, the distance of the vertical projection 
of P above the ground line is equal to the height of the 
point P above H. 



24 WORKING DRAWINGS 

Thus, m Fig 4, the position of P will be fully repre- 
sented by the projections pp' on a perpendicular to the 
ground line (AB) drawn through the point c at any 
given distance from B. 

82. Corollary. — If either projection of a point be 
given, the other projection must lie on the line drawn 
perpendicular to the ground line and passing through 
the given projection. 

83. Problem. — To determine the point in space when 
its projections are given. Pigs 2 or 4. 

If through the given projections p and p' perpendicu- 
lars be erected, they will lie in the same plane and inter- 
sect each other at the point P, thus fixing its position. 

In figure 4, the perpendicular through p and p', after 
revolution, w^ould be parallel, and therefore not intersect; 
but it must be remembered that Y is only supjjosed to be 
revolved into H for convenience in representing the 
parts of the drawing, whereas it is in reality vertical, and 
must always be imagined to be so when reading a draw- 
ing. The pupil should not neglect to practice indicating 
the position of the points in space, by holding a pencil 
point in the proper place, always remembering that Y is 
supposed to stand vertical. The position of P will then 
be indicated by holding a point three-fourths of an inch 
above p, which will be where the perpendiculars through 
p and p 7 would intersect in space. 

EXERCISES. 

84. The planes are supposed to be revolved as in Fig. 
4. The points on AB may be assumed at pleasure. 



AND HOW TO USE THEM. 



25 



The + and — signs are used as before (56) to indicate 
directions. 

SCALE — FULL SIZE. 





Distances. 


Points. 


From V. 


From H. 


11 

B 


112 

— A" 


8 // 

12 

— 1" 


F 
G 
II 
I 


10// 

12 

-A 




4 
T2 

±0 





In what angle ? 



This practice should be extended until the student becomes en- 
tirely familiar with the positions of points, for this is the key to 
the successful interpretation of drawings, and too much stress can- 
not be laid upon it, The position of points once thoroughly un- 
derstood, all the rest follows very readily, since lines and surfaces 
are but collections of points, and the same principles are appli- 
cable to them as to points. 

85. Let the angle or plane in wdiich the following 
points are situated be indicated by the student : 



f A> 



>a 



\a> 



*- clmS 



\& 



.ay 



era- 



>at 



I.' 



-aJ 



CV 



W 



CL 



a- 



>a- 



CONVENTIONALITIES. 

86. In making the drawings, to avoid confusion it is 
necessary to adopt certain conventional methods of ex- 
pressing the parts to be represented. Thus, the line 
joining the projections p and p' of a point should 



26 WORKING DRAWINGS 

always be dotted, as ; the protectants of a point, 

as Pp or Pp' (Fig. 3), should be broken : _ ; 

all given lines or the parts of such lines which are visible 

in space should be drawn full and heavy, thus : * »■ ; 

the projections of such lines should be full, but light, as 

; all invisible or auxiliary lines should be drawn 

broken : ,_____; all visible intersections of planes 

are drawn full: ; and all such intersections 

which are invisible should be broken and dotted, as 



87. All sections of solids should be shaded with sec- 
tion lines or cross-hatchings. See Fig. 19 (a). 

88. All shade lines should be drawn heavy. The posi- 
tion of these lines will depend upon the direction of the 
light, the form and position of the object, and the 
position of the point of sight; but since parallel ra,ys 
of light are generally assumed to enter from the upper 
left hand corner at an angle of 45° to the plane of the 
drawing, it has become conventional to make the right 
hand and bottom lines of all jrrojecting surfaces, and 
the left hand and upper lines of all recesses, shade or 
heavy lines. 



CHAPTER IT. 

THE RIGHT LINE. 

89. Principles to be Remembered. — 1. Between two 
points only one straight line can be drawn ; hence, the 
line is given when the position of the points is known. 

90. 2. The intersection of two planes is a straight 
line. 

91. 3. Two parallel' lines "determine" a plane. 

92. 4. If a plane pass through two points it must con- 
tain the straight line joining them. 

93. If two lines be perpendicular to the same plane, 
they must be parallel. 

94. Problem. — To represent a given straight line by 
its projections. Fig. 5. 

Analysis. — We have already seen that the projections 
of a point are found by passing lines through it perpen 
dicular to the planes H and Y. If now this operation 
be applied to all the points in a given straight line, and 
the points at the feet of the perpendiculars be joined by 
lines, they will be the projections of the given line. 

The horizontal projecting perpendiculars, or project- 
ants, being perpendicular to the same plane (II) will be 
parallel (93), and since they all pass through the given 

line they must lie in the same plane (91) ; hence, their 

(27) 



28 WORKING DRAWINGS 

intersections with H will be on a straight line (90). As 
two parallel lines determine a plane, any two points 
may be taken on the given line, and their projections 
being joined by a right line, will give the desired pro- 
jections (89 and 92). 

Construction. — Thus, in Fig. 5, let P and Q be two 
points whose co-ordinates are given, and let PQ be the 
given right line joining them. Through P let fall the 
protectant p'p upon H, and through Q the protectants 
Qq and Qq' upon H and Y. Then pq will be the hori- 
zontal, and p'q' the vertical projection of the given line. 

95. Remark. — -If a line be produced so as to pierce 
either plane of projection, the point in which it pierces 
the plane will be one point of its projection on that 
plane. For P and p' are the same point. Also, if a 
point be on a right line, its projections must be on the 
projections of the line. 

96. Problem. — To assume a point on a line whose 
projections are given. 

Assume a point on either projection of the line, and 
through it draw a line perpendicular to the ground line, 
until it intersects the other projection of the given line. 
This will be the other projection of the point on the 
line. Thus, in Fig. 5 (a), assume any point, as q, on the 
horizontal projection of the line, and draw qq' perpen- 
dicular to the ground line until it intersects pV at q'. 
These will be the projections of Q on P R. 

The line may be situated in any one of the four angles, 
but we will discuss here only its possible positions in 
the first. 



AND HOW TO USE THEM. 29 

POSITIONS OF RIGHT LINES IN SPACE. 

9?. 1. A line in the first angle may be oblique to the 
ground line — that is, to both H and V. 

2. It may be oblique to H and parallel to Y. 

3. It may be parallel to H and oblique to Y. 

4. It may be parallel to both H and Y — that is, to 
the ground line. 

5. It may be perpendicular to H. 

6. It may be perpendicular to Y. 

T. It may be perpendicular to their intersection. 

8. It imiy lie on the horizontal plane, and be either 
parallel, perpendicular, or oblique to the ground line ; or s 

9. It may have similar positions in the vertical plane. 
Note. — Let the pupil hold a pointer or straight edge 

in their several positions, to aid his imagination in con- 
ceiving of their actual places in space. The wall and 
floor of a room may be taken as the planes Y and H, or 
the leaves of a book turned at right angles. For such 
practice, model planes will be found very useful. These 
various positions of a line in space are represented in 
Figs. 5 to 12 inclusive. There are two diagrams given 
for each case ; the first being the isometric projection 
of the line as seen in space; the second, or figure 
marked (a), being the ordinary orthographic projection 
after the revolution of Y. 

98. Observations. — It will be seen from the first 
case, that the projections pV — pr, Fig. 5, are inclined 
to the ground line, and in revolving Y about AB the 
point p' simply describes a quarter of a circumference 



30 WORKING DRAWINGS 

about the axis, remaining at the same distance from it, 
so that the angle which pV makes with the ground line 
is not changed ; and the position of p r in H is not af- 
fected by revolving Y; hence, in (a) the projections pV 
— pr will represent the same line PR as in Fig. 5. We 
conclude, then, that when a line is oblique to both planes 
of projection, its projections will both be oblique to the 
ground line ; and since -p and p', r and r ; must be in the 
perpendiculars to the ground line, the two projections of 
the same line must be found immediately above and be- 
low each other. 

99. In Case 2, Figs. 6 and (a), it should be observed 
that since the line is parallel to Y, its vertical pro- 
jection pV must be parallel to the line itself r and 
its horizontal projection parallel to the ground line. 
The angle which the line PR makes with H is equal to 
that which pV makes with the ground line, for u two 
plane angles are equal when their sides are parallel and 
lie in the same direction." 

100. In Case 3, Figs. 7 and (a), the conditions are 
similar to the above, only interchanging Y and H. In 
general, therefore, if a line be parallel to either plane 
of projection, its projection on that plane will be parallel 
to the line itself, and its other projection will be parallel 
to the ground line. Also, the angle which the line makes 
with the plane to which it is not parallel, will equal that 
made by its projection on the other plane with the 
ground line. 

101. In case 4, Fig. 8, both projections must be par- 



AND HOW TO USE THEM. 31 

allel to the ground line ; for, a line which is parallel to 
two planes at the same time is parallel to their intersec- 
tion. Also, if through a line which is parallel to a 
given plane any planes be passed, the intersections of 
these planes with the given plane will be straight lines 
parallel to the given line. 

102. Case 5, Fig. 9. ]f a line be perpendicular to H, 
it is evident that all of its points will have the same 
horizontal projection p, and that the lower extremity of 
the line, or Q, will coincide with p, or the three points, 
Q, p, and q, all form one point. The vertical projection 
p'q' will be parallel and equal to the given line, for the 
reason given in case 4, and also because PQp'q' is a 
rectangle. 

103. Case 6, Fig. 10, corresponds in principles to the 
one just given ; hence, we see that if a line be perpen- 
dicular to either H orY, its projection on H or Y will 
be a point, and on Y or H, a right line perpendicular to 
the ground line, and equal in length to the given line. 

104. Case T, Fig. 11. When the line is perpendicular 
to the ground line, it must lie in a plane perpendicular 
to the ground line. This figure shows two positions of 
such a line, one of which intersects the ground line, and 
the other piercing' H and Y. In Fig. (a) the horizontal 
and vertical projections form one line, and hence in this 
case the position of the line in space is only determined 
when the projections of two of its points are known. 
Or the projection may be made upon the side plane to 
which the line is parallel, when it becomes the same as 
Case 2, by substituting S for Y. Hence, 



32 WORKING DRAWINGS 

When a line is perpendicular to the ground line, both 
of its projections must be perpendicular to that line at 
the same point. 

105. Case 8, Fig. 12. If a line lie in either plane of 
projection in any position, it is its own projection on that 
plane, and its projection on the other plane must be in 
the ground line, between the perpendiculars to it through 
the extremities of the given line. 

Case 9 is similar to the above. 

106. Problem. — To determine the position of the line 
in space, having given its projections. 

Analysis. — Since the projectants of any two points 
of the given line are parallel, they form a plane which is 
perpendicular to the plane of projection and passes 
through the given line. This plane, as Prr', Fig. 5, is 
the vertical projecting plane of the line, and its inter- 
section pV with Y is evidently the vertical projection 
of the given line. The other plane, Ppr, is called the 
horizontal projecting plane,* and its intersection with 
H is the horizontal projection of the given line; hence, 
if we pass planes through the given projections of a line, 
perpendicular to the planes H and Y, their intersections 
will be the required line P R in space. 

EXERCISES. 

107. The following lines are given by their projec- 
tions after Y has been revolved into H, and the stu- 
dent is requested to indicate the position of the line in 

*Note the difference between the horizontal projecting -plane, and the 
horizontal plane of projection (H ). 



AND HOW TO USE THEM. 



33 



space as well as to describe it, assuming Y to stand at 
right angles to H. 

108. These lines are all given in the first angle, and 
may be designated as the lines A, B, etc. 





i^^^ 


— X_ 


1 1 


J 


VI 








\, 




i / i 






, X 








' i / 






\ y i 






^\a- 


7 


t 


d/ 


: & \ 


/S kj 








h. 




109. Problem. — To find the 'point in which a given 
right line pierces either plane of projection. Fig. 5. 

Principles. — It has already been shown (95) that the 
point in which a line pierces either plane of projection is 
a point of its projection on that plane; that if a point 
is in either H or V, its projection on Y or H mast be 
on the ground line (57); that if a point is on a line, its 
projection must be on the projections of the line (95) ; 
and finally, a point can only be on two lines at the same 
time when it is at their intersection. 



9# 



34 WORKING DRAWINGS 

Problem. — To find the point in which a line pierces 
H. The required point is in H, and also on the line ; 
hence, its vertical projection must be on the ground 
line as well as on the vertical projection of the line, or at 
their- intersection (r 7 ) Fig. 5, and the horizontal project- 
ion of the same point must be on the perpendicular to 
the ground line through this vertical projection, and also 
on the horizontal projection of the line, or where they 
intersect at r. We have then the following : 

110. Rule. — To find the point in which a straight line 
pierces H, produce its vertical projection until it inter- 
sects the ground line ; at this point erect a perpendicular 
until it cuts the horizontal* projection of the line, pro- 
duced if necessary. This last intersection will be the 
required point. 

This same rule will apply to a point in Y, by trans- 
posing the words vertical and horizontal, and is true for 
any angle. 

111. If the line is parallel to either plane, it is evident 
it cannot pierce it, for its projection on the other will be 
parallel to the ground line, and hence will not intersect it. 

Construction. — Let P Q, Fig. 5 (a), be the given line. 
To find where it pierces H, produce the vertical projec- 
tion p'q' until it cuts the ground line at i J ; erect the 
perpendicular r'r until it intersects the horizontal pro- 
jection of the line pq produced at r. Hence, r is the 
desired point. 

EXERCISE. 

Let the student find the points in which the lines 



AND HOW TO USE THEM. 35 

ABC, given in (108), pierce H and Y, and also assume 
other lines for this purpose. 

TWO LINES WHICH INTERSECT. 

112. Theorem. — If two lines intersect at a point, their 
projections must pass through the projections of the 
point. See Figs. 13 and (a). 

Analysis. — Two intersecting lines may be either ob- 
lique or at right angles to each other ; in either case they 
must have one common point through which, if the pro- 
tectants be drawn, they will lie in the projecting planes 
of the two lines, and hence be at their intersection. But 
these projecting planes intersect H and Y in the projec- 
tions of the lines, which must therefore pass through the 
projections of the common point. 

Construction. — Let PQ and RS be the two given 
lines intersecting at (not shown in (a)). If now the 
horizontal projecting planes p'pq and s'sr be passed 
through them, they will intersect each other in the verti- 
cal line Oo, which is the projectant of the point of inter- 
section, and which pierces H at o, the intersection of the 
horizontal projections of the lines pq and rs. In the 
same way it will appear that Oo', the vertical projectant, 
pierces V at o', where the vertical projections of the 
given lines intersect ; but these points oo' are the pro- 
jections of the same point 0, in which the given lines 
intersect. The converse of this is also true, viz. : 

113. If the projections of two lines pass through the 
projections of the same point, the lines must pass 
through the point in space. 



36 WORKING DRAWINGS 

114. Corollary. — It also follows that to draw a line 
through a given point on another line, it is only neces- 
sary to assume a point on that line (96), and draw the 
projections of the second line through that point ; or, 

115. To draw two or more lines through a given point, 
draw their projections at pleasure through the projec- 
tions of the point. See the figures. 

The student should also show that p' and s' are the 
points in which the given lines pierce Y, and q and r f 
the points in which they pierce H, by applying the rule 
given in 110. 

exercises (After the revolution of V into H). 

116. Let the co-ordinates of a point be S= — \ n ', Y= 
1", and H= 1", natural scale. 

1. Draw through the above point a line parallel to the 
ground line also two others, one perpendicular to H, the 
other to Y. 

2. Draw a line through the same point perpendicular 
to the ground line. 

3. Draw a line parallel to Y, and making an angle of 
45° with H (there may be two such lines, one on either 
side). 

4. Assume a point on the line parallel to the ground 
line, and drop perpendiculars from it to Y and H, thus 
completing the parallelopipedon. 

5. Assume any point in space, and draw any two 
oblique lines through it. 

6. Find the points in which these lines pierce the 
planes of projection. 



CHAPTEK III. 

PLANES. 

11 T. A plane is determined when it passes through 
three given points ; through a right line and a point ; 
through two right lines which intersect ; through two 
parallel lines ; through a point and perpendicular to a 
given line ; through a point and parallel to a plane, and 
by many other conditions. Since two intersecting 
straight lines determine a plane, if the points be found 
in which they pierce H and Y, these points will lie in 
the plane of the two lines, and the straight lines joining 
them (one of the lines being in H, the other in Y), will 
also lie in the plane of the given lines ; hence, they are 
the intersections of the plane of the given lines with H 
and Y. 

118. These intersections are called the traces of the 
plane on H and Y. 

119. Theorem. — The traces of an oblique plane must 
intersect each other, if at all, on the ground line. For 
the horizontal trace is a line of H, and it is not parallel 
to the ground line, which is also a line in H. Since 
then these two lines lie in the same plane, and are not 
parallel, they must intersect at some point. For the 

same reason the vertical trace and gruond line must 

(37) 



38 WORKING DRAWINGS 

intersect at some point. But the two given traces also 
lie in the same oblique plane, and are not parallel; 
hence, they must intersect each other as well as the 
ground line; therefore, these three lines must have one 
point in common, which is where the traces meet the 
ground line. There can be but one such point, since a 
straight line (AB) can pierce a plane but once. 

The two traces and the ground line form the three 
edges of a triangle or pyramid whose vertex is their in- 
tersection. 

120. Problem. — To represent a plane in any position 
by its traces. 

A plane, like a line, may have a great variety of posi- 
tions in space. The following are some of those in the 
first angle : 

1. It may be oblique to both H and Y or to the 
ground line, as p'Tr, Fig. 13. 

2. It may be perpendicular to H but inclined to V, as 
s'sr, Figs. 13 and (a), in which case the vertical trace is 
perpendicular to the ground line, and the horizontal trace 
makes an angle Tsr with the ground line equal to that 
between the planes— p'pq is another instance of a simi- 
lar position. 

3. It may be perpendicular to both H and V at any 
point of AB, in which case both traces will be perpen- 
dicular to AB at the same point, and will form one 
straight line. As in Figs. 14 and (a). 

4. It may be parallel to H, in which case it can only 
intersect V in a line parallel to the ground line. It will 



AND HOW TO USE THEM. 39 

then have but one trace parallel to AB, and at a distance 
from it equal to that between the planes as shown in 
Figs. 15 and (a). 

Or it may be parallel to V, when its trace will be on H 
and parallel to AB. Figs. 16 and (a). 

These traces are only distinguishable from the projec- 
tions of lines parallel to AB by the letters T, T, t', t, 
used to designate them. 

5. And finally the plane may be parallel to the ground 
line and yet intersect both H and V, as in Figs. IT 
and (a). 

In this case the intersection of the given with the 
side plane will determine the angle of inclination. 

The student should indicate these positions by placing 
a card in the angle of the model planes of projection. 

APPLICATIONS. 

121. Problem. — To represent a simple geometrical 
solid, as a rectangular block of wood, stone, or brick, by 
its projections. 

To represent a brick, the dimensions of which are 8^-x 
4x2^ inches, by a drawing made to a scale of |- full size, 
the dimensions of the drawing must be one-eighth of the 
above figures, or l^g-x^-x^ inches ; hence, if we assume 
the brick to lie with its edges parallel to H and Y. and 
touching them, the length may be laid off on the ground 
line ; and from its ends lay off the height above and the 
breadth below it. See Fig. 18, (a). From this drawing a 
model ^ the full size may readily be made. Bj changing 
the scale, the model may be made of any desired size. 



40 WORKING DRAWINGS 

122. The portion of a drawing made on the vertical 
plane or wall is called the elevation; that on the hori- 
zontal plane or ground, the ground plan, or simply the 
plan, and those on the side planes, ma}' be either end 

ELEVATIONS Or VERTICAL SECTIONS. 

The great simplicity of Fig. 18 (a), over the obliqie 
perspective of the same solid, as given in Fig. 18, is at 
once apparent, and yet the latter gives all the informa- 
tion contained in the former; for it will be seen at once 
that the points P and Q are in planes perpendicular to 
the ground line, to which the edge PQ is parallel ; that 
the plane of the upper face is parallel to II, and hence 
has but one trace, p'q', whilst the front face is parallel 
to Y and intersects H in the line pq, etc. 

123. Problem. — To represent a box of the same dimen- 
sions as above. 

If we wish to construct a box, as in Fig. 19, it would 
be better to show the cross section, although the size of 
the interior might be expressed in Fig. 18 (a) by draw- 
ing dotted lines at the proper distance from the projec- 
tions of the outer edges, as in 19 (a), in which the cross 
section is also shown. 

In this case the thickness of the sides of the box is 
assumed to be \ of an inch, which will be represented in 
the drawing by T y, the scale being as before, ^ full size. 

The inner and outer surfaces of the box are in reality 
two rectangular prisms, but the edges of the inner one, 
being concealed by the outer prism, are drawn dotted. 

Sections. — The plane cutting out a right section may 



AND HOW TO USE THEM. 41 

be taken perpendicular to the edge at any point. In the 
drawing it is represented at t'Tt, and its intersection 
with the solid material of the box is shaded with section 
lines. This section is projected upon the side plane, 
which is then revolved to the right about BC, until it 
coincides with Y. This brings the section on t't on the 
right side of Fig. 19 (a) and above the ground line. 

124. Thus the true sizes of the pattern are obtained 
from this drawing for all the various parts of the box. 
It is simply the unfolding of three adjacent faces, and is 
evidently one-half of the whole surface. The amount of 
material required can therefore be readily estimated. 

The student should compute the required amount. 

125. The dimensions are marked on the drawing for 
full size, and the scale should be marked to correspond. 
The advantage of numbering the subdivisions of the first 
unit to the left will become apparent in using the divid- 
ers to measure distances. 

This is the simplest position in which to represent a 
rectangular figure, but there are many others which may 
be necessar}\ These will be explained subsequently. 

126. Problem. — To represent an assemblage of rectan- 
gular blocks of the same or different sizes. Figs. 20 
and (a). 

The same principles are applied in projecting a num- 
ber of parts as in representing any unit ; thus in Fig. 20, 
the bricks or blocks in a wall are shown in precisely the 
same manner as the single block in Fig. 18, but on a 
smaller scale. Here the scale is ^ full size, or 24 



42 WORKING DRAWINGS 

inches to 1 inch ; that is, two feet to one inch, or one 
foot to one-half an inch. In drawing the scale, there- 
fore, one-half an inch should be marked to represent 
one foot. 

127. Problem. — To represent a square mortise and 
tenon joint. Fig. 21. 

This joint consists of a rectangular projection called a 
tenon, cut upon the end of a piece of timber, which is 
made to fit a corresponding recess, called a mortise, cut 
on the face of another piece, which is to be framed or 
joined to the first. 

Thus, in Fig. 21, is shown the tenon T, and under it 
the mortise M. The same parts are shown in Fig. 21 
(a) in plan, side and end elevation. In this case, the 
lower block, called a "sill" does not rest against V, but 
stands out from it a distance equal to m'n. If then m'C 
be the axis of revolution for the side plane, the revolved 
position of n will be at n", and of m at rn /; . These 
points revolve in H, which is perpendicular to the axis, 
and remaining at distances from it equal to m m 7 and 
n m 7 , the radii of their respective circumferences. So of 
any other point, as R, which revolves about the point r' 1? 
in the axis. Its projection upon the side plane is at R 1? 
and this point when revolved will be found at r^ , at a 
distance r^ r" from the axis equal to the radius m'r^. 
The plane in which the point moves being perpendicular 
to the axis at the centre r' 17 will have but one trace, 
which will pass through that point and be parallel to the 
ground line, and it is in this trace that r£ is found. 
Hence this 



AND HOW TO USE THEM. 43 

128. Rule. — To find the revolved position of the pro- 
jections of a point on a side plane. Through the vertical 
projection of the point draw a right line perpendicular 
to the vertical trace of S, and lay off on this line from 
the trace a distance equal to the distance of the horizon- 
tal projection of the point from the ground line. 

Such are a few of the large number of applications that 
may be made of the principles already given. The 
pupil can readily extend them to a variety of other cases. 



CHAPTEK IV. 

PROBLEMS RELATING TO LINES AND 
PLANES. 

129. Problem. — To find the true length of an oblique 
line joining two given points in space. Case I. 

It will be seen from the preceding problems that the 
projections of a line are only equal to the line itself when 
the line is parallel to the planes of projection. In 
order, then, to find the true length of any line given by 
its projections, it will be necessary to revolve the line 
either into or parallel to one of the planes of projection. 
Its projection on that plane will then be equal to the 
line itself. 

Analysis. — If the line does not pierce either H or Y 
within the limits of the drawing, imagine either of its 
projecting planes to pass through it, and revolve this 
plane, containing the given line, about its trace as an 
axis, until it coincides with the plane of projection. 
The revolved position of the line will then give its true 
length. 

Construction.— Let PQ, Figs. 22 and (a), be any ob- 
lique line. Through it pass its horizontal projecting 
plane Pq. It cuts H in pq, which is the projection of 

the line. Now revolve the plane Pq about pq and find 

(44) 



AND HOW TO MAKE AND USE THEM. 45 

the revolved position of P and Q in H, hy the rule al- 
ready given (64). The line p^q", will be the true length 
of PQ. The perspective Fig. 22 does not give the true 
length. 

A similar construction, using p'q' as an axis, would 
have given the length upon Y. 

130. Another method. — Or, the plane Pq may be re- 
volved about the perpendicular Pp as an axis until par- 
allel to Y, when PQ will be projected upon Y.in its true 
size. 

This position is not shown in Fig. 22, to avoid confu- 
sion, but is given in (a), where o' is the centre of the 
circumference described by Q moving in a plane parallel 
to H, and whose vertical trace is, therefore, the line o r q\ 
The radius being pq, when the plane Pq is revolved until 
parallel to Y, the point q will be found at q^", and p r q 7/ 
will be the required length. 

Note. — It will be seen that the length of an oblique 
line is always greater than that of its projections. 

131. Case II.— When the line pierces either H or Y. 
Analysis. — The point in which it pierces the plane is 

a point of its projection on that plane, and hence is on 
the axis. It is only necessary, therefore, to revolve the 
other end of the line into the plane of projection, and 
join this point with that in the axis for the true length. 

Let the pupil assume such a line, and determine its 
length. 

132. Problem. — To assume a line in any oblique 
plane. 



46 WORKING DRAWINGS 

Case 1. When the line is oblique to both planes of pro- 
jection. 

Analysis. — Since the straight line joining any two 
points of a plane must be a line of that plane, and since 
the traces of the given plane are also lines of the plane, if 
airy point in one of these traces be joined to any point in 
the other by a right line, that line must lie in the plane. 
The line is represented by its projections, as in Figs. 13 
and (a). The converse of this proposition is also true. 

Construction. — (Same Fig.) Let p be any point in 
the vertical trace of the plane p'Tr, and q be any point 
in the horizontal trace of the same plane, the line P Q 
joining them must lie in the plane. Its vertical projec- 
tion is p'q 7 , and its horizontal pq. 

133. Case 2. When the line is parallel to either plane 
of projection. 

Analysis. — If a line be drawn through a point of a 
plane parallel to a line of that plane, it must lie in the 
plane ; hence, if a line be drawn through any point of 
one trace parallel to the other, it will lie in the given 
plane, and be parallel to H or Y. 

Construction. — Figs. 23 and (a). Let t'Tt be the 
given plane. Through any assumed point, as n, in the 
horizontal trace tT, draw a line, as NM parallel to Tt'. 
This line being parallel to IV, will be parallel to the 
plane in which it lies, and therefore its projection on Y 
will be parallel to the line itself, NM, or to its parallel 
line Tt'. Its horizontal projection will be parallel to the 
ground line (99). 



AND HOW TO MAKE AND USE THEM. 47 

134. Case 3. If the line be perpendicular to the 
ground line. 

In this case its projections must be perpendicular to 
the ground line (104). In Figs. 23 and (a) the line is 
PQ. It intersects MN, which is also a line of tTt' in 0, 
as will be seen from its projections. From this case it 
also appears that if any point of either trace be joined 
with the points of any line of the given plane by straight 
lines, these lines will lie in the plane. 

135. Problem. — To pass a plane through three given 
points. 

Analysis. — If through any two of the points right 
lines be drawn intersecting at the third, these lines will 
line in the plane of the points. The traces of this plane 
will be found by producing the lines until they pierce 
H and V, and joining the points thus found. Should 
the lines not pierce the planes of projection within 
the limits of the drawing, assume any other points 
(96) on the two lines joining the given points, and join 
them by a line which will give points of the required 
traces. 

Application.— Figs. 13 and (a). Let R N be the 
three given points ; join II and by a straight line, and 
find the points r and s' in which it pierces H and Y. 
These will be points in the required traces. Join and 
ST, and find the points p- and q as before. The lines r q 
and p' s' will be the horizontal and vertical traces of the 
plane of the points. If the construction be correct, 
these traces will intersect each other, if at all, on the 
ground line at T. 



48 WORKING DRAWINGS 

136. Problem. — To pass a plane through two linen 
which intersect or are parallel. 

The general solution of this problem is the same as 
that of the preceding after the points have been joined 
by right lines. See figs. 13 and (a). 

In case the parallel lines are parallel to the ground 
line, or to either plane of projection, it is only neces- 
sary to assume points on the given lines, and join them 
by other lines which will pierce the planes H or Y. 

137. Theorem. — If two or more lines are parallel, 
their projections must be parallel ; for the horizontal 
projections are determined by passing planes through 
the lines and perpendicular to H. These projecting 
planes must therefore be parallel, and hence their inter- 
sections with H, or the horizontal projections of the 
lines, are parallel to each other.* For the same reason 
the vertical projections are parallel to each other. 

138. Problem. — To pass a plane through one given 
line parallel to another. 

Analysis. — If a line be drawn through any point of 
the given line (114) parallel to the other (137), these 
intersecting lines will determine the required plane. 
The traces are found as in (135) or (109). Let the 
pupil construct the problem. 

139. Problem. — To pass a plane through a right line 
and a point. 

Analysis — Let the point be joined to any assumed 

*"If two parallel planes be intersected by a third plane, the 
lines of intersection will be parallel." 



AND HOW TO MAKE AND USE THEM. 49 

point on the given line, and the plane of the intersecting 
lines will be the one required. The traces are found as 
in (136). 

TWO PLANES. 

140. Problem. To find the intersection of two oblique 
planes ivhose traces intersect. 

Analysis. — The required intersection must be a 
straight line lying in both planes, but we have seen 
(132) that if a line lies in a plane it must pierce the 
planes of projection, if at all, in some point of the 
traces of the planes. As the line of intersection is in 
both planes, it can only pierce the planes H and Y in the 
points where the traces of the given planes intersect 
each other ; hence if a line be drawn, by its projec- 
tions, between these points, it will be the required inter- 
section. 

Construction. — Let sSs' and tTt', Figs. 24 and (a), 
represent the two given oblique planes. The vertical 
traces intersect at p r , and the horizontal at r, the line P 
R, joining them, is the desired intersection; its projec- 
tions are pr and p f r f . 

141. Problem. — To find the angle which the line of 
intersection P II makes with either plane of projection, 
as H. 

Analysis. — Revolve the line about its projection on 
H as an axis, into the horizontal plane, when the angle 
between the revolved position and the axis will be the 
one required. 

Construction. — Since the point r, Fig. 24 (a), is on 



50 WORKING DRAWINGS 

the axis rp, it remains fixed. The revolved position of 
p' will be found, as in (64), at p". Hence p"r is the re- 
volved position of the line, and p"rp is the angle. 

142. Remark. — The angle which a line makes with 
any plane is the same as th'at which it makes with its 
projection on that plane. 

143. Special Cases. — If the intersecting planes are 
perpendicular to H onlv, the vertical traces must be 
perpendicular to the ground line (120, case 2), but the 
horizontal traces will be oblique ; and since the planes 
intersect, their traces must meet in some point, through 
which the line of intersection must pass. As the planes 
are perpendicular to H, this line must also be perpendic- 
ular to H or parallel to the vertical traces, hence its 
position is fully determined, as it passes through a given 
point and is parallel to a given line. 

Construction. — In Fig 25 and {a) the planes sSs' and 
tlV are perpendicular to H, their vertical traces, perpen- 
dicular to the ground line, the line of intersection PR 
passes through the point r, in which the horizontal 
traces intersect, and is parallel to Y, or to the vertical 
traces. The pupil should construct the figure for the 
case in which the planes are perpendicular to Y, which 
is just the reverse of this. 

144. Problem. — When the intersecting planes are par- 
allel to the ground line. 

Analysis. — In this case the intersection will be par- 
allel to the ground line.* Its position is best de- 

* A line can only be parallel to two planes at the same time when it 
is parallel to their intersection. 



AND HOW TO MAKE AND USE THEM. 51 

termined by cutting the two given planes by a side or 
section plane, finding their traces upon this plane, and 
through their point of intersection, drawing a line par- 
allel to the ground line. 

Construction. — In figures 20 and (a), the planes tt'P 
and ss'P are cut by the side plane in the lines tt' and ss', 
which intersect at R, the line through which, parallel to 
the ground line, is the required line. In Fig. (a) the 
side plane is revolved about its vertical trace, and the 
vertical projection r" is found in its revolved position; 
its true projections r and r' are easily determined, and 
the required line P R is then known. 

145. Problem. — To find the intersection of two ob- 
lique planes token the traces do not intersect within the 
limits of the drawing. In this case let us assume that 
the horizontal traces do not intersect. If we suppose a 
third plane to be passed parallel to the vertical plane, it 
will cut the given oblique planes in two lines which will 
be parallel to the vertical traces' of those planes 1 (99), 
and these lines must cut each other at some point of the 
required line of intersection. 2 A second point of the 
line may be found in the same manner, and thus the line 
will be determined by points. 

Construction. — Let r'Tt and r'Ss be the given oblique 
planes whose horizontal traces do not intersect within 

x If an oblique plane cut two parallel planes, the lines of interection 
will be parallel ; and the projections of parallel lines are parallel. 

2 If two lines lie in the same plane and are not parallel, they must in- 
tersect. 



52 WORKING DRAWINGS 

the drawing. Pass a plane whose H trace is n m, par- 
allel to V. It will cut the given planes in the lines PM 
and PN, the vertical projections of which are p'm' and 
p'n' parallel to the vertical traces and intersecting at p', 
the vertical projection of P, one of the required points. 
Since P is in a vertical plane, its H projection must be 
found in the horizontal trace mn of that plane, and also 
in the perpendicular to the ground line through p', 
hence at p. In like manner q' and q are found, and the 
line joining Q and P to them should pass through the 
point in which the vertical traces intersect. If the ver- 
tical traces do not intersect, the construction is the 
same. 

146. Problem. — To find the point in which a given 
right line pierces any oblique plane. 

Analysis. — If through the given line any auxiliary 
plane be passed, it will cut the given plane, if at all, in 
a line which must meet the given line at the required 
point.* Hence if a plane be passed through the given 
line so as to cut the given plane, the line of intersection 
of these planes will meet the given line in the required 
point, which will be found where the projections of the 
lines intersect (113). 

Construction. — Let tTt',' Figs. 28 and (a), be the 
given plane, and P Q the line. For simplicity, pass 
through P Q its horizontal projecting plane pSt'. Its 

*For every right line of a plane must pierce any other plane to 
which the line is not parallel in the common intersection of the two 
planes. 



AND HOW TO MAKE AND USE THEM. 53 

traces intersect tTV in the points m and t f ; hence the 
line mt' is the intersection of the two planes. This line 
intersects P Q in O, which is therefore the desired point. 

141. Problem. — To find a point in a plane when 
either of its projections is given. 

Analysis. — Through the given projection of the point 
erect a perpendicular to the plane of projection. This 
must pass through the required point. Hence if the 
point in which the perpendicular pierces the given plane 
be found, as in the last problem, it will be the one 
required. 

Construction. — Let o, Figs. 28 and (a), be the given 
projection of the point, and Oo the perpendicular 
through it. Pass any plane, as pSt', through the per- 
pendicular. It intersects the given plane tlV in the 
line t'ni, which cuts the perpendicular at or o' in Fig. 
(a), which is therefore the required point. 

PRACTICAL APPLICATIONS. 

148. Problem. — To construct a simple dove-cote, clog- 
kennel or similar structure out of boards , without a 
frame. 

Remarks. — The first step in this problem is to decide 
upon the form and dimensions. This requires an exer- 
cise of the imagination, and knowledge of the habits 
and size of the proposed occupants of the house. The 
particular style is a matter of taste, and may be varied 
to suit the idea of the designer. In this ease the 
building is intended for a medium-sized dog. Its 
dimensions will be assumed, therefore, at 24 inches in 



54 WORKING DRAWINGS 

length, 18 in breadth, and 34 in height, measured out- 
side. The boards are to be one inch thick. The next 
step will be to select a convenient scale, so as to repre- 
sent all the necessary parts upon a page or paper of 
limited dimensions. In this case it is one-sixteenth the 
full size, which will just enable the three projections to 
be placed upon the Plate (viii), Fig. 29. 

Description of Parts. — The structure consists of 7 
parts, viz. : 2 ends or gables, 2 sides, 2 roof boards, and 
1 base plate. This latter may be allowed to project one 
or more inches all around. The doors and windows 
should be cut out before the parts are put together. 
The dormer windows in the roof are merely imitations, 
made of wedge-shaped blocks covered with half-inch 
boards. Before beginning any work, the amount of 
material required should always be determined, and a 
bill made out in the following form : 

BILL OF MATERIAL. 

1 Base -board, 28x22= 616 square inches. 

2 Sides, 22X18= 792 
2 Gables, 32X18=1152 
2 Roof-boards, 30X23=1380 



144)3940 



27 -f square feet. 

As the material cut out for doors and gables will fur- 
nish blocks for dormers, etc., it will not be necessary to 
make a large allowance for waste ; so that thirty feet of 
lumber will be sufficient. 

The cost of materials will then be, for 



AND HOW TO MAKE AND USE THEM. 55 

30 feet 1 -inch boards @ $0.05=$1.50 
2 pounds assorted nails (a} $.05= .10 



$1.60 
Construction. — Having the materials and tools, the 
pupil can proceed to erect the structure, without detailed 
instruction from a text-book, as to the manner of using 
them. A few failures and a little patience will prove to 
be the best instructors. The angles of the bevels on the 
edges of the roof and sides are given by the drawing, 
Fig. 29, in the end elevation. The eaves of the roof 
may be left square. 

In the same manner any similar structure composed 
of olanes and straight lines may be erected. 



A MANUAL 



OF 



Engineering Specifications § Contracts, 

(REVISED EDITION) 

Designed as a Text-Book and Work of Reference 

for all who may be engaged in the Theory or 

Practice of Engineering. 

By LEWIS M. HAUPT, C. E., 

Prof, of Civil Engineerinc;, To^vne Scientific 

School, University of Pennsylvania, 

Philadelphia,. 

Octavo, Cloth, Illustrated, - J*rice $3,00. 



TESTIMONIALS. 

Bowdoin College, 
Brunswick, Maine, Feb. 8th, 1878. 

Prof. Lewis M. Haupt : My Pear Sir : I have received to-day 
your work upon Engineering Specifications and Contracts. You 
have certainly supplied one of the missing links between the theory 
and the practice of Engineering. The work will save an immense 
amount of time to the practitioner, and will be of great service to the 
student, by calling his attention to many matters of detail which he 
would look in vain for elsewhere. I shall give it to ray students as 
a text-book. Very truly yours, GEORGE L. VOSE, 

Prof. Civil Engineering. 

Thayee. School of Engineering, 
Connected with Dartmouth College, 
Hanover, N. H., Feb. 18th, 1878. 
My Dear Prof. Haupt : I acknowledge with many thanks the 
receipt of a copy of your work, "Engineering Specifications and Con- 
tracts/' So far as I can judge from such examination as my time 
has yet allowed me to give it, the book is admirable in its complete- 
ness and adaptability to the purposes for which it was designed. I 
have adopted it as a text-book for mv students. 

ROBERT FLETCHER, 

Prof, of Civil Engineering. 



Feb. 26th, 1878. 
My Dear Professor : I write a line merely to state that I am 
more and more pleased with your book as I proceed with the perusal 
of it. It supplies a great fond of information of the utmost value, 
which either is not accessible to a young engineer, or is so scattered 
that it would become available onlv after laborious research. 

Very truly yours, ' ROBERT FLETCHER, 

Prof, of Engineering. 



Washington, D. C., Feb. 26&, 1878. 
Your " Manual" is a most valuable acquisition to the student and 
to the engineer, and I feel that with such a book ray past experience 
would authorize rae to undertake the practice of the professsion. 

E. D. CUTTS, 
Assistant in Charge of Goedetic Operations, U. S. Coast Survey. 



" The title of this work at once justifies its introduction to the pub- 
lic as calculated to fill the existing and acknowledged vacancy ; 
while the name of the author inspires confidence in the manner of 
execution of so admirable a project. An examination of the book 
does not disappoint the expectations thus raised. The subjects of 
drawings, estimates and measurements, specifications, advertisements, 
bids and contracts, are considered from the standpoints of the engin- 
eer, the contractor, the man of business, and the 'lawyer, with excel- 
lent practical judgment, and each chapter is provided also with 
appended questions, by the use of which the book becomes a manual 
of instruction. Many valuable tables for application in practice, a 
glossary of technical terms, and a full index, complete the usefulness 
of the work, which we heartily recommend to both instructors and 
constructors." — Engineering and Mining Journal. 



" Civil engineers and contractors on public works or on any exten- 
sive mechanical structure, will welcome the handsome new volume 
called 'A Manual of Engineering Specifications and Contracts/ by 
Professor Lewis M. Haupt, of the Chair of Civil Engineering in the 
Towne Scientific School of the University of Pennsylvania, who has 
had the advantage of practical experience as a civil engineer. The 
work, which is a technical one in the essential parts, is also one that 
will be found of value to young engineers, and to contractors who 
may wish to learn how to secure their own rights as well as to know 
their Own obligations. The book is the first of the kind ever pub- 
lished in the United States, and it seems to be as thorough and com- 
plete as such a book could possibly be made. Although handsomely 
printed and bound, and containing many illustrations and much tab- 
ular work, this most excellent manual is retailed at the low price of 
three dollars." — Philadelphia Evening Bulletin. 



" This is an instructive manual of an uncommon kind, but of the 
highest importance to the young engineer. It covers that branch 
of engineering education which can be satisfactorily learned from a 
book, without other aid. Even experienced engineers cannot aflbrd 
to trust to memory alone in drawing specifications, and will find a 
carefully prepared book of valuable service." — Von Nostrand's 
"Engineering Magazine/' N. Y. City. 

And many others to the same effect. 

J. M. STODDART & CO., 

727 Chestnut St., Philadelphia. 



FIGS. 1-2. 



THE PLANES 




POINTS 




Scale in inones. Full Size . 



■i i i i -i — r 



FIGS. 3-4. 



REVOLUTION of POINTS and PLANES 




3 <b> 




!V T 




c 






■it r ; 




/ 




P s 



Pig. 4 



Scale V, 



-I— J — C_L 



FIGS. 5-12 



PROJECTIONS ofaFUGHT LINE 

(a) 




'2* Angle 
Exercises 



FIGS. 13-17. t. \JL 



PLANES. 

Fig.13, / 

i " / (a) i '« 

/A/ 

LA 


l' 

Fig.14. 


/ \ 

'! \ 








i 






T' T' 






\ 

Fig.I5. \ (a) 

A B 


Pig. 


\ 

it; 

i 


. 


• 

A 




\ 

B 


• 

l-i-J .17. 

A B 


i 




(a 


' 






r 















FIGS. 18-21. u 



APPLICATIONS. 
-K Fi£.ia. ^ (ai 



^ 



Elevation \ 



Elev.ititm 



Ground Plan 



in u 11 to e a 




p 


|r' * 




1 — 


Elevation i 


im^^^m. 




|| Section. 










_L 




7-/, r i 

Planj 


(a) 



Fig. 20. 



(a> 




FIGS. 22-2^ 



" 



GENERAL PROPOSITIONS. 




b. 



FIGS. 25-27 



INTERSECTIONS of PLANES 



8 


^~— jF 




t' 






(a.) 






3 




v' 




r 














Fifi.26 




FIGS. 28-29 




Fi£28 



APPLICATIONS 
Fi£ 29. 



ft ft ft 



U U D 



Side -Elevation.-. 




PL 



£ml Elevation 



ScaLe I6inL.to 1 hi.. 



FIG. 30 




LIBRARY OF CONGRESS^ 

019 970 435 7 



av Text Book and Work of 
£& Reference 

FOR ALL WHO MAY BE ENGAGED IN THE 

Theory or Practice of Engineering, 



drawings, 
Estimates, 
cifications, 
Advertisements, 
Proposals, 

Contracts. 

DESIGNED AS A 




BY 



LEWIS M. HAUPT, C.E., 

PROFESSOR OF CIVIL ENGINEERING, TOWNE SCIENTIFIC SCHOOL, 
UNIVERSITY OF PENNSYLVANIA. 



PHILADELPHIA: 

Published by- J. M. STODDART & CO., 727. Chestnut St. 

1 88 j. 



